President Obama has used virtually every public appearance in recent weeks to demand an immediate congressional vote on his jobs plan.

On Tuesday evening, he’ll get it, as the Democratic-controlled Senate plans to hold a procedural vote on whether to allow the $447 million American Jobs Act to move forward.

But the vote, which could be the only one Congress will hold on the package as a whole, might do as much to put Democratic senators on the spot as it will to highlight the Republican obstructionism that Obama has blamed for blocking consideration of the package.

There is little mystery about the vote’s outcome. Democrats hold 53 seats in the Senate, not enough to overcome unified GOP opposition and muster the 60 votes necessary to break a filibuster and send the package forward.

Even if all Democrats vote for it. And all Democrats are unlikely to back it. Eyeing Obama’s dipping popularity and a mixed reception in polls for the jobs package, several Democratic senators from swing states have shied away from embracing it.

Obama’s demand for a vote has put senators facing difficult reelection bids next year in a tough position: Vote for a bill that is probably headed for defeat, and they will be painted by opponents as too closely aligned with Obama. But vote with Republicans against it, and they will undermine Obama’s claim that congressional Republicans have been the obstacle to his attempts to improve the economy.

That is likely to be a dynamic that will prove problematic for Obama and Senate leaders in coming months.

Democratic leaders will spend Tuesday urging wavering senators that it is in their interest to vote “yes,” and aides say they are confident that most of their caucus will back the plan.

The package includes spending for school construction and transportation infrastructure; payroll tax cuts for workers and small businesses; and tax credits for businesses that hire veterans.

Last week, Democratic leaders in the Senate revised the bill to try to make it more palatable to their members — and a more potent political tool against Republicans who oppose it.

Instead of paying for the package by ending government subsidies to the oil and gas industry and limiting tax deductions for those making more than $250,000 a year, as Obama had proposed, they suggested a 5.6 percent surtax on millionaires.

With the revision, Senate leaders think the bill presents a sharper choice: Support spending on education, transportation and veterans, or protect the interests of the very wealthy.

“Republicans will be hard-pressed to explain why they’d allow teachers and firefighters to be laid off rather than have millionaires and billionaires pay their fair share,” Sen. Charles E. Schumer (D-N.Y.) said last week.

The changes helped bring some hesitant Democrats into the fold. Sen. Mary Landrieu, who represents Louisiana and its offshore oil industry, called the new way to pay for the bill “a key step in the right direction” and said she was eager to move it forward.

Sen. Robert P. Casey Jr. (D-Pa.), who will face a tough reelection battle next year and had said he thinks Congress should consider Obama’s plan in pieces, said late last week that the millionaires’ tax is a “far better approach” to pay for the ideas.

“Now that we’re in the fiscal crisis that we’re in and the jobs crisis we’re in, we have to take action to create jobs,” he said. “This proposal is one way to effectuate the priority of creating jobs.”

But a number of other Democrats have said they oppose the plan or declined to indicate how they will vote.

Sen. Jon Tester (D-Mont.) told a newspaper in his home state that he fears cutting the payroll tax could hurt Social Security, which is paid for from the tax. And he said he thinks that offering businesses a tax credit to hire the long-term unemployed, as Obama has proposed, is not the best way to create jobs.

“There are things I like in this proposal, but they are outweighed by the things I just can’t accept right now,” he told Lee newspapers.

The Democratic hesitancy has allowed Republicans to pin the plan’s likely defeat on Obama’s allies.

In the weekly Republican address Saturday, Sen. John Thune (S.D.) said that Obama’s plan is “so flawed that Senate Democrats have rejected it and are rewriting it, not to grow jobs, but to improve their political standing.”

“It’s nothing but a rehash of the same failed ideas he’s already tried, combined with a huge tax increase,” he said.

The president has been largely silent about murmurings of discontent in his party.

Instead, in appearances at schools and factories in eight states, he has trained his rhetorical firepower on Republicans in Congress, including House leaders who have refused to schedule a vote on the package.

“If the Republicans in Congress think they have a better plan for creating jobs right now, they should prove it,” Obama said Saturday in his weekly video address.

On Monday, he sent supporters a tweet about the Senate’s vote: “Make sure your Republican legislator knows you support it.”